Properties

Label 1800.1.u.b
Level 1800
Weight 1
Character orbit 1800.u
Analytic conductor 0.898
Analytic rank 0
Dimension 4
Projective image \(D_{4}\)
CM disc. -24
Inner twists 8

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Newspace parameters

Level: \( N \) = \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 1800.u (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.898317022739\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{4}\)
Projective field Galois closure of 4.0.9000.2

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q\) \( -\zeta_{8}^{3} q^{2} \) \( -\zeta_{8}^{2} q^{4} \) \( + ( 1 + \zeta_{8}^{2} ) q^{7} \) \( -\zeta_{8} q^{8} \) \(+O(q^{10})\) \( q\) \( -\zeta_{8}^{3} q^{2} \) \( -\zeta_{8}^{2} q^{4} \) \( + ( 1 + \zeta_{8}^{2} ) q^{7} \) \( -\zeta_{8} q^{8} \) \( + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{11} \) \( + ( \zeta_{8} - \zeta_{8}^{3} ) q^{14} \) \(- q^{16}\) \( + ( -1 - \zeta_{8}^{2} ) q^{22} \) \( + ( 1 - \zeta_{8}^{2} ) q^{28} \) \( + ( \zeta_{8} - \zeta_{8}^{3} ) q^{29} \) \( + \zeta_{8}^{3} q^{32} \) \( + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{44} \) \( + \zeta_{8}^{2} q^{49} \) \( + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{56} \) \( + ( 1 - \zeta_{8}^{2} ) q^{58} \) \( + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{59} \) \( + \zeta_{8}^{2} q^{64} \) \( + ( -1 + \zeta_{8}^{2} ) q^{73} \) \( -2 \zeta_{8}^{3} q^{77} \) \( -2 \zeta_{8} q^{83} \) \( + ( -1 + \zeta_{8}^{2} ) q^{88} \) \( + ( 1 + \zeta_{8}^{2} ) q^{97} \) \( + \zeta_{8} q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut +\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut +\mathstrut 4q^{7} \) \(\mathstrut -\mathstrut 4q^{16} \) \(\mathstrut -\mathstrut 4q^{22} \) \(\mathstrut +\mathstrut 4q^{28} \) \(\mathstrut +\mathstrut 4q^{58} \) \(\mathstrut -\mathstrut 4q^{73} \) \(\mathstrut -\mathstrut 4q^{88} \) \(\mathstrut +\mathstrut 4q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1001\) \(1351\)
\(\chi(n)\) \(\zeta_{8}^{2}\) \(-1\) \(1\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
757.1
−0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i 0 1.00000i 0 0 1.00000 + 1.00000i 0.707107 + 0.707107i 0 0
757.2 0.707107 0.707107i 0 1.00000i 0 0 1.00000 + 1.00000i −0.707107 0.707107i 0 0
1693.1 −0.707107 0.707107i 0 1.00000i 0 0 1.00000 1.00000i 0.707107 0.707107i 0 0
1693.2 0.707107 + 0.707107i 0 1.00000i 0 0 1.00000 1.00000i −0.707107 + 0.707107i 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
24.h Odd 1 CM by \(\Q(\sqrt{-6}) \) yes
3.b Odd 1 yes
5.c Odd 1 yes
8.b Even 1 yes
15.e Even 1 yes
40.i Odd 1 yes
120.w Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{7}^{2} \) \(\mathstrut -\mathstrut 2 T_{7} \) \(\mathstrut +\mathstrut 2 \) acting on \(S_{1}^{\mathrm{new}}(1800, [\chi])\).